Fractional Cointegration and Aggregate Money Demand Functions
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CORE Metadata, citation and similar papers at core.ac.uk Provided by Brunel University Research Archive FRACTIONAL COINTEGRATION AND AGGREGATE MONEY DEMAND FUNCTIONS Guglielmo Maria Caporale Brunel University, London Luis A. Gil-Alana University of Navarre Abstract This paper examines aggregate money demand relationships in five industrial countries by employing a two-step strategy for testing the null hypothesis of no cointegration against alternatives which are fractionally cointegrated. Fractional cointegration would imply that, although there exists a long-run relationship, the equilibrium errors exhibit slow reversion to zero, i.e. that the error correction term possesses long memory, and hence deviations from equilibrium are highly persistent. It is found that the null hypothesis of no cointegration cannot be rejected for Japan. By contrast, there is some evidence of fractional cointegration for the remaining countries, i.e., Germany, Canada, the US, and the UK (where, however, the negative income elasticity which is found is not theory-consistent). Consequently, it appears that money targeting might be the appropriate policy framework for monetary authorities in the first three countries, but not in Japan or in the UK. Keywords: Money Demand, Velocity, Fractional Integration, Fractional Cointegration JEL Classification: C32, C22, E41 We are grateful to an anonymous referee for useful comments, and to Mario Cerrato for research assistance. The second named author also gratefully acknowledges financial support from the Gobierno de Navarra, (“Ayudas de Formación y Desarrollo”), Spain. Corresponding author: Professor Guglielmo Maria Caporale, Brunel Business School, Brunel University, Uxbridge, Middlesex UB8 3PH, UK. Tel. +44 (0)1895 203 327. Fax: +44 (0)1895 269 770. E-mail: [email protected] 1. Introduction The existence of a stable long-run relationship linking real money balances to real income and interest rates has been extensively investigated, as it would provide support for money targeting as a policy strategy for monetary authorities. Whilst early empirical studies seemed to suggest that a log- linear equation of this kind exhibited stability (see, e.g., Goldfeld, 1973), subsequently it became apparent that the relationship had broken down, both in the US and in the UK, possibly as a result of policy changes. Goldfeld and Sichel (1990) concluded that the existing empirical models suggested instability in the money demand function (see Goodhart, 1989). The claim that a stable money demand exists, more specifically that real M1, real income and short-term interest rates are cointegrated, or, alternatively, that velocity is a stationary variable (the two statements being equivalent, as long as interest rates are I(0)), has resurfaced in some recent studies. A typical example is the work of Hoffman, Rasche and Tieslau (1995) and Hoffman and Rasche (1996), who, using the Johansen (1988, 1991) procedure, find cointegrated long-run demand functions for narrowly defined money (M1) in five industrial countries including the UK. This result is in contrast to some of the stylised facts about money demand. Consider, for instance, the case of the UK (see Goodhart, 1989). Existing empirical studies generally reach the conclusion that monetary aggregates are not cointegrated with nominal income1; however, cointegration can be achieved by adding other variables, such as wealth, financial innovation variables or cumulated interest rates, to a standard money demand function (see, e.g., Hall et al. 1990, Hendry and Ericsson, 1991, 1993). This is because velocity is highly trended, and only by including a similar effect is it possible to build a well-balanced equation. There is a very clear downward trend until 1981, after which year an increasingly important component of M1 began to earn interest and so M1 began to behave more like broad money. Then this is followed by a clear 1 reversal and a strong upward trend. Even if one ignores this break in definition it is not clear that velocity is a stationary process. Caporale et al (2001) argue that the test procedure in Hoffman et al (1995) and Hoffman and Rasche (1996) is crucially affected by the specification of the VAR, and show that when an adequate VAR can be achieved the assumption of cointegration is no longer supported. Furthermore, where there are important structural breaks which are not adequately dealt with it may be impossible to achieve correct inference from this procedure. In this paper we provide further evidence on whether or not there exist stable money demand functions by using fractional integration and cointegration techniques. The motivation is the following. Earlier studies rely on standard cointegration analysis, and either conclude that all variables are I(1) and the equilibrium errors are an I(0) process, which is not persistent (e.g., Hoffman et al, 1995), or, alternatively, that they are I(1) (e.g., Friedman and Kuttner, 1992). However, the discrete options I(1) and I(0) offered by this type of analysis are rather restrictive, as adjustment to equilibrium might take a longer time than suggested by standard cointegration tests if, in fact, the equilibrium errors respond more slowly to shocks, which results in highly persistent deviations from equilibrium. In other words, real money balances, real income and nominal interest rates might be tied together through a fractionally integrated I(d)-type process such that the equilibrium errors are I(d), with d < 1, and exhibit slow reversion to zero. Consequently, it is possible that a failure to identify a long-run equilibrium consistent with money demand theory simply reflects the adoption of a classical cointegration framework. Therefore, we adopt a testing procedure which allows for the possibility of a long-memory cointegrating relationship, and which enables us to gain a better understanding of money demand. Specifically, we test the null hypothesis of no cointegration against alternatives which are fractionally cointegrated using the two-step strategy presented in Gil-Alana (2003) and Caporale 1 For a survey of the evidence, see Goodhart (1989) and also Temperton (1991). 2 and Gil-Alana (2004). By applying this methodology and using data for five major industrial countries (namely, Canada, US, Japan, Germany, and UK), we find mixed results. In particular, the null hypothesis of no cointegration cannot be rejected for Japan, but there is some evidence of fractional cointegration for Germany, Canada, the US, and the UK (where, however, the income elasticity is found to be negative). In such cases, although there exists a long-run relationship, the error correction term may possess long memory, which means that deviations from equilibrium are highly persistent. Hence, our findings are to some extent, but not wholly, consistent with the conclusions reached by other studies of the behaviour of monetary aggregates in the industrial countries (see, e.g., Caporale et al, 2001), and suggest that money targeting might be the appropriate policy framework for monetary authorities in some countries (where a stable relationship appears to exist), though not in some others (where instability prevails, or elasticities are not theory-consistent).2 The layout of the paper is the following. Section 2 briefly describes the concepts of fractional integration and cointegration, and the procedure adopted for testing the null hypothesis of no cointegration against fractionally cointegrated alternatives. The empirical results are presented in Section 3. Finally, Section 4 offers some concluding remarks. 2. Testing for fractional integration and cointegration As already mentioned, in studies relying on standard cointegration analysis the equilibrium errors are restricted to be an I(0) process, which is not persistent. However, it might be the case that the equilibrium errors respond more slowly to shocks, which results in highly persistent deviations from equilibrium. The testing procedure described below allows for the possibility of such long- memory cointegrating relationships. 2 Some recent papers investigate possible instabilities as well as nonlinearities using the smooth transition regression techniques developed by, e.g., Granger and Terasvirta (1993) and Lin and Terasvirta (1994), and tend to find stable linear relationships (see, e.g., Wolters and Lütkepohl, 1997, and Wolters, Teräsvirta and Lütkepohl, 1998). 3 For the purpose of the present paper, we define an I(0) process {ut, t = 0, ±1, …..}, as a covariance stationary process with spectral density that is positive and finite at zero frequency. In this context, an I(d) process, {xt, t = 0, ±1, ….}, is defined by: d (1 – L) xt = ut, t = 1, 2, … , (1) 3 xt = 0 t ≤ 0, (2) where L is the lag operator and ut is I(0). The macroeconomic literature focuses on the cases d = 0 and d = 1 (see, e.g., Nelson and Plosser, 1982), whereas we define (1 – L)d for all real d by: ∞ d k ⎛d ⎞ k d (d −1) 2 d (d −1)(d − 2) 3 (1 − L) = ∑(−1) ⎜ ⎟ L = 1 − d L + L − L + ... k = 0 ⎝k ⎠ 2 6 The process ut in (1) could be a stationary and invertible ARMA sequence, with an exponentially decaying autocovariance function. This property can be said to characterise a “weakly autocorrelated” process. When d = 0, xt = ut, so a “weakly autocorrelated” xt is allowed for. When d = 1, xt has a unit root, while for a general integer d, xt has d unit roots. For 0 < d < 0.5, xt is still covariance stationary, but its lag-j autocovariance γj decreases very slowly, like the 2d-1 power law j as j → ∞, and so the γj are non-summable. The distinction between I(d) processes with different values of d is also important from an economic point of view: if a variable is an I(d) process with d ∈ [0.5, 1), it will be covariance nonstationary but mean-reverting, since an innovation will have no permanent effect on its value. This is in contrast to an I(1) process, which will be both covariance nonstationary and non-mean-reverting, in which case the effect of an innovation will persist forever.